2.1 solving equations graphically

2.1 SOLVING EQUATIONS GRAPHICALLY

Objectives:1. Solve equations using the intersect

method.2. Solve equations using the x-intercept

method.

Example #1aSolve using the intersection method.

x2 3x 5

x

3 x 2A)

Enter both equations into separate lines of the Y = screen. To enter absolute value equations, go to MATH NUM 1:abs(Then graph both lines an adjust the window if necessary.


x2 3x 5

x

3 x 2A)The solution to this equation occurs where the two equations intersect each other. This can be found on the calculator by pressing 2nd TRACE (CALC) 5:intersect.

The calculator then asks you three questions: First Curve? Second Curve? Guess?


x2 3x 5

x

3 x 2A)The first two questions are designed to have you identify which two lines to find the intersection of. This is useful if more than two lines are present on the screen. By pressing enter for the first question, the cursor automatically goes to the second curve on the screen. Otherwise the curves can be selected by using the arrow keys and then pressing enter. It is also important to note that the intersection must be present onscreen for this to work.

The last question is necessary when the two selected lines intersect each other more than once. In this situation a “guess” of the solution is necessary so the calculator knows which intersection you are trying to find. The guess must be an x-value.

For this problem, x = 2 was guessed although any number would work since it only intersected once. The approximate solution is then shown onscreen as the x-coordinate. Remember that we are solving for x, so the y-coordinate should be ignored.

Example #1b Solve using the intersection method.

B) 0.1x4 5

0.1x 3 0.1x 12

This time, the entire graph does not show up on the screen. Although one intersection is shown, it appears that the lines will intersect each other twice. To get both intersections to fit onscreen at the same time, the window needs to be adjusted.


B) 0.1x4 5

0.1x 3 0.1x 12

By pressing the WINDOW key, the window size can be adjusted. By setting the Ymin to 0 and the Ymax to 20, both intersections will show onscreen. This can be a bit of a trial and error process.

Alternatively, ZOOM 0:ZoomFit will usually automatically resize the window so that both curves “fit” onscreen at the same time, but it does not always work well and may still need zoomed out further.


B) 0.1x4 5

0.1x 3 0.1x 12

This time the intersection function will need to be performed twice on the calculator.

To get the left intersection, −3 would be a good guess, to get the right intersection, 4 would be a good guess.

Example #2aSolve using the intercept method.

x 4 2x2 x 5 4

0 x5 x4 2x2 4First rewrite the equation with all terms on one side.

This new equation will be entered into the Y = screen using only a single line as opposed to the dual lines of the intersection method.

A)


x 4 2x2 x 5 4A)

The solution(s) to the intercept method is where the line crosses the x-axis. This can be found using 2nd TRACE (CALC) 2:zero.

As with the intersect method, the calculator asks a series of questions: Left Bound? Right Bound? Guess?

Because a curve my cross the x-axis multiple times, it is necessary for you to pinpoint for the calculator which solution you are trying to find by putting boundaries on where to look for the solution and by making a guess.


B) x 4 2x 2 x 5 1

1 2 3–1–2–3 x

1

2

3

4

5

6

7

8

9

–1

–2

–3

–4

–5

–6

–7

–8

–9

y

When selecting the left bound, you want to choose a number that is on the left side of where the line crosses the x-axis. A good choice for this graph would be either −3 or − 2. Even decimals could be chosen such as − 1.7 as long as it is clear that the choice is on the left side of the zero.

For the right bound, a selection of − 1 or 0 would be good choices.

When making a guess, make sure that you guess a number in between your left and right bound. For instance if −2 & −1 are chosen for the bounds, −1.3 would be a good choice. Selecting 0 which is outside the bounds will produce an error.


B) x 4 2x 2 x 5 1The guessing of the solution is necessary if two zeros are inside the boundaries you selected, so the calculator can differentiate which one you want to find, but the calculator will still make you guess regardless of the actual number of zeros.

As with the intersection method, the solution is given as the x-coordinate.

Example #3Solving by solving

x4 3x3 2x2 1 0

f(x) 0 f(x) 0

Entering an equation with a square root into the graphing calculator will show a graph that does not touch the x-axis. This is an error on part of the calculator as the square root of 0 does equal 0 so a solution should exist. (**Note**: When using intersect method include the square root.)


x4 3x3 2x2 1 0

f(x) 0 f(x) 0

By graphing the equation without the square root symbol, a different graph will appear, but the function shares the same zeros. In other words, this graph crosses the x-axis in the exact same places the other function should have touched the x-axis.


x4 3x3 2x2 1 0

f(x) 0 f(x) 0

By performing the zero function on the calculator twice, both solutions to the original equation can then be found.

For the left zero, -1 & 0 were chosen as the boundaries, for the right zero, 1 & 3 where chosen for the boundaries.

Example #4Solving . f(x)

g(x) 0

2x 2 x 38x2 7x 4

0

Rational functions can be very difficult to read off of a TI calculator screen. First of all, to enter them into the calculator the numerator and denominator must be in separate parentheses. TI-84 calculators will also do a better job representing them than TI-83+ calculators, but both still aren’t the best.


g(x) 0

2x 2 x 38x2 7x 4

0

Using alternative graphing technology, better graphs can be found, but they are not necessary to find the zeros of the function.

Places where the denominator equals zero are undefined in the function, only the numerator is necessary to find the zeros of the function.

If the graph of the numerator is superimposed onto the original function you can see that the zeros occur in the same places.

1 2 3 4 5–1–2–3–4–5 x

1

2

3

4

5

–1

–2

–3

–4

–5

y

1 2 3 4 5–1–2–3–4–5 x

1

2

3

4

5

–1

–2

–3

–4

–5

y


g(x) 0

2x 2 x 38x2 7x 4

0

Here is what the graph looks like with only the numerator entered into the Y = screen.


g(x) 0

2x 2 x 38x2 7x 4

0

By performing the zero function twice, both solutions can be found.

–2 & 0 were chosen for the left solution and 1 & 2 for the right solution.

Example #5Equal Populations

According to data from the U.S. Bureau of the Census, the approximate population y (in thousands) of Baltimore, Maryland and Austin, Texas between 1970 and 2003 are given by

Baltimore:

Austin:

Where x represents the number of years past 1970. In what year did the two cities have the same population?

y 0.0114x 3 0.6203x2 16.573x 905.04

y 0.0076x3 0.543x2 3.4649x 255.63


Baltimore:

Austin:

Graph both equations in the Y = screen. Since they are given to us as separate equations, the intersection method would work best.

y 0.0114x 3 0.6203x2 16.573x 905.04

y 0.0076x3 0.543x2 3.4649x 255.63

Neither graph shows up on the screen with the default zoom. By pressing ZOOM 0:ZoomFit, both graphs will show up on the screen but the intersection is not showing. If you zoom out (ZOOM 3:Zoom Out Enter) the intersection will now show onscreen.


After performing the intersection, the solution is shown as follows:

The original question was:

“Where x represents the number of years past 1970. In what year did the two cities have the same population?”

If we add 30.6 years to 1970, the two cities will have the same population in about July of 2000.

2.1 solving equations graphically

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